Narrow Results

In this paper we consider the thin film approximation of a 1D scalar conservation law with strictly convex flux. We prove that the sequence of approximate solutions converges to the unique Kružkov solution.

We develop a version of Haar and Holmgren methods which applies to discontinuous solutions of nonlinear hyperbolic systems and allows us to control the L¹ distance between two entropy solutions. The main difficulty is to cope with linear hyperbolic systems with discontinuous coefficients. Our main observation is that, while entropy solutions contain compressive shocks only, the averaged matrix associated with two such solutions has compressive or undercompressive shocks, but no rarefaction-shocks — which are recognized as a source for non-uniqueness and instability. Our Haar–Holmgren-type method rests on the geometry associated with the averaged matrix and takes into account adjoint problems and wave cancellations along generalized characteristics. It generalizes the method proposed earlier by LeFloch et al. for genuinely nonlinear systems. In the present paper, we cover solutions with small total variation and a class of systems with general flux that need not be genuinely nonlinear and includes for instance fluid dynamics equations. We prove that solutions generated by Glimm or front tracking schemes depend continuously in the L¹ norm upon their initial data, by exhibiting an L¹ functional controlling the distance between two solutions.

We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W:={𝟙}_{(-\infty ,0]}(\cdot )exp(\cdot )\ast \rho$ satisfy an Oleĭnik-type entropy condition. More precisely, under different sets of assumptions on the velocity function $V$, we prove that $W$ satisfies a one-sided Lipschitz condition and that $V^{\prime }(W)W{\partial }_{x}W$ satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation.

The authors give the first convergence proof for the Lax-Friedrichs finite difference scheme for non-convex genuinely nonlinear scalar conservation laws of the form

In usual a method to discretize partial differential equation is required to have some consistency with the solution to original differential equation. When we solve hyperbolic conservation laws numerically by difference approximation, it is very important to know how consistent with the entropy solution the difference approximation or numerical result is. In the case of scalar conservation law, we know that a difference approximation converges to the entropy solution if it satisfies some condition which is described in terms of numerical viscosity coefficients. (See [1, 2, 10, 13].) In these works, the consistency with entropy condition is an important subject and is discussed through the numerical entropy inequality or cell entropy inequality, which is a discrete analogue of the entropy inequality that is employed in the definition of entropy condition by Lax. (See [6, 12].) On the other hand, the results on convergence may lack practical information on the numerical behavior of difference approximation, where the numerical behavior means the behavior of numerical result which is calculated at some finite values of difference increments (Δx, Δt etc.). But information on numerical behavior is important when we are interested in the quality of numerical computation.

We here discuss the consistency with entropy condition from the viewpoint of Oleǐnik's E-condition. Although the discussion is not applied so widely as that based on the numerical entropy inequality, the result gives more information on the numerical behavior of difference approximations. It helps us discuss the quality of numerical computation.